Abstract: We report on the long term X-ray monitoring of the outburst decay of the low
magnetic field magnetar SGR 0418+5729, using all the available X-ray data
obtained with RXTE, SWIFT, Chandra, and XMM-Newton observations, from the
discovery of the source in June 2009, up to August 2012. The timing analysis
allowed us to obtain the first measurement of the period derivative of SGR
0418+5729: \dot{P}=4(1)x10^{-15} s/s, significant at ~3.5 sigma confidence
level. This leads to a surface dipolar magnetic field of B_dip ~6x 10^{12} G.
This measurement confirms SGR 0418+5729 as the lowest magnetic field magnetar.
Following the flux and spectral evolution from the beginning of the outburst up
to ~1200 days, we observe a gradual cooling of the tiny hot spot responsible
for the X-ray emission, from a temperature of ~0.9 to 0.3 keV. Simultaneously,
the X-ray flux decreased by about 3 orders of magnitude: from about
1.4x10^{-11} to 1.2x10^{-14} erg/s/cm^2 . Deep radio, millimeter, optical and
gamma-ray observations did not detect the source counterpart, implying
stringent limits on its multi-band emission, as well as constraints on the
presence of a fossil disk. By modeling the magneto-thermal secular evolution of
SGR 0418+5729, we infer a realistic age of ~550 kyr, and a dipolar magnetic
field at birth of ~10^{14} G. The outburst characteristics suggest the presence
of a thin twisted bundle with a small heated spot at its base. The bundle
untwisted in the first few months following the outburst, while the hot spot
decreases in temperature and size. We estimate the outburst rate of low
magnetic field magnetars to be about one per year per galaxy, and we briefly
discuss the consequences of such result in several other astrophysical
contexts.